Gravitational fields and Newton's law of universal gravitation: VCE Physics Unit 3

What this dot point is asking

VCAA wants you to describe gravity using a field model, apply Newton's law of universal gravitation, calculate field strength $g$, and find changes in gravitational potential energy both in a uniform field ($mg\Delta h$) and in a non-uniform field (the area under the $F$ vs $r$ graph).

The answer

The field model

A gravitational field surrounds every mass. A second mass placed in the field experiences a force toward the first. The field is a vector at each point, drawn with field lines that point in the direction of the force on a positive test mass.

• Near a planet's surface the field is approximately uniform (parallel field lines, $g \approx 9.8$ N/kg downward).
• At large distances the field is radial (lines pointing inward toward the planet) and the strength falls as $1/r^2$.

Newton's law of universal gravitation

Every pair of masses attracts with a force:

where $G = 6.67 \times 10^{-11}$ N m squared per kg squared and $r$ is the centre-to-centre distance.

Field strength

The gravitational field strength $g$ at a point is the force per unit mass:

It is also numerically equal to the acceleration of a freely falling object (because $F = mg$ and $F = ma$ together give $a = g$). Units are N/kg or m/s squared.

At Earth's surface, $g = 9.8$ N/kg. The field strength falls with altitude as $1/r^2$ where $r$ is measured from Earth's centre.

Gravitational potential energy in a uniform field

Close to a planet's surface, the field is approximately uniform. The change in gravitational potential energy when an object of mass $m$ is raised through height $\Delta h$ is:

This is the work done against gravity.

Gravitational potential energy in a non-uniform field

Far from a planet's surface, the field varies with distance and the equation $mg\Delta h$ no longer applies. The change in potential energy between two distances $r_1$ and $r_2$ equals the work done against the gravitational force, which is the area under the $F$ vs $r$ graph between those points.

VCAA expects you to use the graphical area for non-uniform fields. The analytic equivalent is:

for moving from $r_1$ to $r_2 > r_1$ (positive when moving outward, because work is done against gravity).

Field lines and energy: a quick visual rule

• Field lines point in the direction of the force on a positive test mass (always toward the planet).
• A mass moved with the field (downward) loses potential energy.
• A mass moved against the field (outward) gains potential energy.

Examples in context

Example 1. ANU Mt Stromlo satellite orbit determination. Mt Stromlo's satellite-laser-ranging station tracks the $9$ tonne LAGEOS satellite at altitude $5900$ km, orbital radius $r = 6378 + 5900 = 12\,280$ km. Newton's law gives gravitational field strength at this radius: $g = GM/r^2 = 6.674 \times 10^{-11} \times 5.97 \times 10^{24}/(1.228 \times 10^7)^2 = 2.64$ m s$^{-2}$. Centripetal balance gives orbital speed $v = \sqrt{gr} = \sqrt{2.64 \times 1.228 \times 10^7} = 5694$ m s$^{-1}$, and period $T = 2\pi r/v = 2\pi \times 1.228 \times 10^7/5694 = 1.35 \times 10^4$ s = $3.77$ hours, matching the observed orbital period.

Example 2. Snowy 2.0 pumped-hydro gravitational PE storage. Snowy 2.0 lifts water $700$ m from Talbingo (lower) to Tantangara (upper). Treating $g$ as uniform at $9.8$ m s$^{-2}$ over this $700$ m altitude range, gravitational PE stored per kg is $\Delta U = mgh = 9.8 \times 700 = 6860$ J, or $1.91$ kWh per tonne. Tantangara's $254$ GL ($2.54 \times 10^{11}$ kg) thus stores $2.54 \times 10^{11} \times 6860 = 1.74 \times 10^{15}$ J or $485$ GWh of potential energy. Released through turbines, this powers $2200$ MW for $\approx 220$ hours, providing the seasonal balance for renewable variability.

Try this

Q1. State Newton's law of universal gravitation and define each symbol. [2 marks]

• Cue. $F = Gm_1 m_2/r^2$. $G$ is the gravitational constant; $m_1, m_2$ are interacting masses; $r$ is centre-to-centre distance.

Q2. Earth-orbit radius $R_E = 6.378 \times 10^6$ m, $M_E = 5.97 \times 10^{24}$ kg. Calculate (a) gravitational field strength at surface, and (b) at altitude $400$ km (ISS orbit). [4 marks]

• Cue. (a) $GM/R^2 = 9.80$ m s$^{-2}$. (b) $r = 6.778 \times 10^6$, $g = 8.68$ m s$^{-2}$.

Q3. Refer to Snowy 2.0 PE storage. (a) Calculate the energy stored per kg lifted $700$ m. (b) Determine the total stored energy in $254$ GL of water. (c) Explain why $g$ may be treated as uniform over this height range. [2+3+2 marks]

• Cue. (a) $6860$ J. (b) $1.74 \times 10^{15}$ J. (c) Altitude change of $700$ m is small compared with Earth radius, so $g$ varies by less than $0.02\%$.